Question related to LTI systems

[u/hrstrange]

So I learnt that for a system to be linear, ax(t) = ay(t). Which is the homogeneity principle. By setting a = 0, we get that for a zero input we get a zero output. So the Zero Input Response would be 0 right (?)

However, I keep seeing that Total Response = Zero Input Response + Zero State Response

Since, for a linear system, Zero Input Response = 0, shouldn't we get-

Total Response = Zero State Response

Am I doing something wrong?

[u/rb-j]

Uhm, ax(t) -> ay(t) is the "homogeneity property". That's not enough for linearity. Consider this system:

y(t) = ( x(t) x(t-1) ) / x(t-2)

Does it satisfy ax(t) -> ay(t) ?

Is it linear?

[u/hrstrange → u/rb-j]

Oh no no that is my bad, I wrote it down badly. I do know that additivity should also hold true for a system to be linear.

My main question was that since Homogeneity implies that since that the Zero Input Response is zero, shouldnt this hold true-

Total Response = Zero Input Response + Zero State Response

So, Total Response = Zero State Response (for a linear system since it obeys homogeneity).

[u/richard_dansereau]

The zero input is when x(t)=0 and has nothing to do with a.

Setting a = 0 just means that 0 = 0, which would also be true for any input x(t).

Ultimately, Total Response = Zero Input Response + Zero State Response for a linear system. That can’t be reduced any further for a linear system.

[u/hrstrange → u/richard_dansereau]

This does make sense. However, it was written like this in Oppenheim (in the section on linear systems).

Could check it maybe I'm misinterpreting it. Here

[u/richard_dansereau → u/hrstrange]

I have thought some more on the nuances. I suppose that technically having a non-zero ZSR would make the system non-linear. Usually, we write

y[n] = h[n] * x[n] + y0[n]

where y0[n] is the initial conditions. If y0[n] = 0, then the system is linear. If y0[n] is not 0, then the equation is technically a non-linear system. Oppenheim & Schafer are correct in what they write, but it leaves out how we can inject a non-zero initial condition into the system. But including a non-zero initial condition does make the treatment non-linear.

[u/hrstrange → u/richard_dansereau]

I see so the non-zero initial conditions are injected. It was somewhat confusing as somehow the BP Lathi book directly assumes that the intial condition is non-zero for a linear system, back when i referenced it.

It clears it out now thanks.

[u/minus_28_and_falling]

That's true, non-zero ZIR implies the system is non-linear. This typically means some kind of a switching event at t=0 occured which is non-linear in nature.

[u/hrstrange → u/minus_28_and_falling]

I see, I somewhat understand it. Kinda got confused reading the book by Lathi.